Theorem 7.3.1 of [1]} developed intrinsic matrix Bernstein concentration inequality. Together with the techniques in Section 3.2 of [2]}, it can be easily extended to compact operators.
Similar results can also be found in Lemma 5 of [3]}.
In the below, \(\mathcal {H}_1\)  and \(\mathcal {H}_2\)  are two Hilbert spaces.
